Consider the $2$DOF mass$-$spring$-$damper

system below:

$k=1000,c=25,m_1=0\mathrm{.}5,m_2=10$

a$.$ Draw a FBD and derive the equation of motion for each mass.

b$.$ Derive the mass, damping, and stiffness matrices of the coupled system.

c$.$ Calculate the undamped natural frequencies and mode shapes of the system Provide a

visual representation of the mode shapes $($sketch or plot$)$